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We show that $ A_N$ satisfies a local scale-free $ \\ell ^{p}$-improving estimate, for $ 3/2 < p \\leq 2$: \\begin{equation*}\n  N ^{-2/p'} \\lVert A_N f \\rVert _{ p'} \\lesssim N ^{-2/p} \\lVert f\\rVert _{\\ell ^{p}}, \\end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =\\frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = \\sup _{N\\geq 1} |A_Nf| $ satisfies a similar sparse"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.05734","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-07-12T13:22:17Z","cross_cats_sorted":[],"title_canon_sha256":"9513b30687f9c379aaf9c1295bd5d672423d31db4d1053e0957ebce68d83cef6","abstract_canon_sha256":"35312ba07eca77a4ff97863450170e41bbc03fe58e3b37cc9b3fcbb14320abb7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:41:00.162525Z","signature_b64":"J98CNRR/9GmztBPP14kvMAyF1/YGWqtZXe4pMZ84wuBodAtc4P2AgPU+HpWp/0sYnonAE8KiVur9f+w/gW2lAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ef835dea1ea190199319e381f07b6dd8a4f3ba4552be906678c04bab7289761","last_reissued_at":"2026-07-05T02:41:00.162080Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:41:00.162080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Averages along the Square Integers: $\\ell^p$ improving and Sparse Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Fan Yang, Michael T Lacey, Rui Han","submitted_at":"2019-07-12T13:22:17Z","abstract_excerpt":"Let $f\\in \\ell^2(\\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\\frac{1}{N}\\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \\ell ^{p}$-improving estimate, for $ 3/2 < p \\leq 2$: \\begin{equation*}\n  N ^{-2/p'} \\lVert A_N f \\rVert _{ p'} \\lesssim N ^{-2/p} \\lVert f\\rVert _{\\ell ^{p}}, \\end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =\\frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = \\sup _{N\\geq 1} |A_Nf| $ satisfies a similar sparse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.05734","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1907.05734/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1907.05734","created_at":"2026-07-05T02:41:00.162149+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.05734v2","created_at":"2026-07-05T02:41:00.162149+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.05734","created_at":"2026-07-05T02:41:00.162149+00:00"},{"alias_kind":"pith_short_12","alias_value":"D34DLXVB5IMQ","created_at":"2026-07-05T02:41:00.162149+00:00"},{"alias_kind":"pith_short_16","alias_value":"D34DLXVB5IMQDGJR","created_at":"2026-07-05T02:41:00.162149+00:00"},{"alias_kind":"pith_short_8","alias_value":"D34DLXVB","created_at":"2026-07-05T02:41:00.162149+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.08893","citing_title":"Sharp $\\ell^p$-Improving Estimates for Fixed-Radius Discrete Spherical Averages","ref_index":9,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W","json":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W.json","graph_json":"https://pith.science/api/pith-number/D34DLXVB5IMQDGJRTY4B6B5W3W/graph.json","events_json":"https://pith.science/api/pith-number/D34DLXVB5IMQDGJRTY4B6B5W3W/events.json","paper":"https://pith.science/paper/D34DLXVB"},"agent_actions":{"view_html":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W","download_json":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W.json","view_paper":"https://pith.science/paper/D34DLXVB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.05734&json=true","fetch_graph":"https://pith.science/api/pith-number/D34DLXVB5IMQDGJRTY4B6B5W3W/graph.json","fetch_events":"https://pith.science/api/pith-number/D34DLXVB5IMQDGJRTY4B6B5W3W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W/action/storage_attestation","attest_author":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W/action/author_attestation","sign_citation":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W/action/citation_signature","submit_replication":"https://pith.science/pith/D34DLXVB5IMQDGJRTY4B6B5W3W/action/replication_record"}},"created_at":"2026-07-05T02:41:00.162149+00:00","updated_at":"2026-07-05T02:41:00.162149+00:00"}